In this activity, students analyze worked examples of equations solved by completing the square. The work allows them to further develop their understanding of the method when used with rational numbers and to notice common errors. Identifying errors in worked examples is an opportunity to attend to precision (MP6) and to analyze and critique the reasoning of others (MP3).

Display the four solutions in the Task Statement for all to see. Select students to share the errors they spotted and their proposed corrections.

To involve more students in the discussion, after each student presents, consider asking students to classify each error by type (not limited to one type per error) and explain their classification. Display these examples of types of errors. Then, ask if students have any additional categories they would like to add to the list:

• Careless errors—for example, writing the wrong number or symbol, repeating the same step twice, leaving out a negative sign, or not following directions.
• Computational errors—for example, adding or subtracting incorrectly or putting a decimal point in the wrong place.
• Gaps in understanding—for example, misunderstanding of the problem or the rules of algebra, applying an ineffective strategy, or choosing the wrong operation or step.
• Lack of precision or incomplete communication—for example, missing steps or explanations, incorrect notation, or forgetting labels or parentheses.

In this activity, students encounter equations that contain rational numbers. They learn that, even though the numbers are not as straightforward as in equations they have previously seen, they can still solve the equations by completing the square. The less friendly numbers and the more elaborate equations make the computations more challenging, so students need to attend to precision (MP6) and mind the steps for rearranging parts of the equations, as well as the signs of the numbers—especially negative numbers.

As students work, they may need to be reminded to rewrite the equation in a different form, isolate a term to make it easier to complete the square, apply the zero product property only when a product is equal to zero, and so on. Make note of common errors or hurdles and address them during whole-class discussion.

Some students may have a difficult time getting started given the increased complexity of these problems. Encourage these students to use completing the square to first solve a more familiar problem, such as , and then to use the structure of the process they followed as a guide to help them solve these more challenging problems.

Direct students' attention to the reference created using Collect and Display. Ask students to share their solutions. Invite students to borrow language from the display as needed, and update the reference to include additional phrases as they respond. (For example, “First, the equation needed to be written in standard form. Then we needed to find half of the coefficient of the linear term.”)

If students found solutions that differ from each other, as time permits, invite these students to describe their thinking to the class, and encourage them to try to identify any errors and determine which solutions are correct. Discuss any common challenges or errors.

• Start with a quadratic expression in standard form on one side of the equal sign and a number (which could be 0) on the other side.
• Take half of the coefficient of the linear term, and square it. The result is the constant term that would make the quadratic expression a perfect square.
• Perform the same operations on both sides of the equal sign so that the constant term on the side with the expression in standard form is a perfect square and makes the expression a perfect square.
• Rewrite the expression (in standard form) as a squared factor.
• Find both values of the factor that give the numbers on the other side of the equal sign when squared.